Fundamental Limits for Sensor-Based Control via the Gibbs Variational Principle

TL;DR

This paper introduces a new lower bound on the performance of feedback controllers with partial observations, using the Gibbs variational principle, applicable to complex dynamics, and demonstrates its effectiveness on a nonlinear tracking problem.

Contribution

It develops a novel, self-consistent lower bound for feedback control performance that accounts for sensor information limits and is computationally feasible.

Findings

The bound accurately predicts the optimal cost in a nonlinear Dubins car tracking task.

The bound tightens with better controller performance and sensor information.

Open-loop bounds are ineffective at low sensor noise levels.

Abstract

Fundamental limits on the performance of feedback controllers are essential for benchmarking algorithms, guiding sensor selection, and certifying task feasibility -- yet few general-purpose tools exist for computing them. Existing information-theoretic approaches overestimate the information a sensor must provide by evaluating it against the uncontrolled system, producing bounds that degrade precisely when feedback is most valuable. We derive a lower bound on the minimum expected cost of any causal feedback controller under partial observations by applying the Gibbs variational principle to the joint path measure over states and observations. The bound applies to nonlinear, nonholonomic, and hybrid dynamics with unbounded costs and admits a self-consistent refinement: any good controller concentrates the state, which limits the information the sensor can extract, which tightens the bound. The resulting fixed-point equation has a unique solution computable by bisection, and we provide conditions under which the free energy minimization is provably convex, yielding a certifiably correct numerical bound. On a nonlinear Dubins car tracking problem, the self-consistent bound captures most of the optimal cost across sensor noise levels, while the open-loop variant is vacuous at low noise.